Browsing by Author "Cockburn, Bernardo"

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    Discontinuous Galerkin Methods with Time-Operators in Their Numerical Traces for Time-Dependent Electromagnetics
    (2022) Cockburn, Bernardo; Du, Shukai; Sanchez, Manuel A.
    We present a new class of discontinuous Galerkin methods for the space discretization of the time-dependent Maxwell equations whose main feature is the use of time derivatives and/or time integrals in the stabilization part of their numerical traces. These numerical traces are chosen in such a way that the resulting semidiscrete schemes exactly conserve a discrete version of the energy. We introduce four model ways of achieving this and show that, when using the mid-point rule to march in time, the fully discrete schemes also conserve the discrete energy. Moreover, we propose a new three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time. The first step consists in transforming the semidiscrete scheme into a Hamiltonian dynamical system. The second step consists in applying a symplectic time-marching method to this dynamical system in order to guarantee that the resulting fully discrete method conserves the discrete energy in time. The third and last step consists in reversing the above-mentioned transformation to rewrite the fully discrete scheme in terms of the original variables.
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    A priori error analysis of new semidiscrete, Hamiltonian HDG methods for the time-dependent Maxwell's equations
    (2023) Cockburn, Bernardo; Du, Shukai; Sanchez, Manuel A.
    We present the first a priori error analysis of a class of space-discretizations by Hybridizable Discontinuous Galerkin (HDG) methods for the time-dependent Maxwell's equations introduced in Sanchez et al. [Comput. Methods Appl. Mech. Eng. 396 (2022) 114969]. The distinctive feature of these discretizations is that they display a discrete version of the Hamiltonian structure of the original Maxwell's equations. This is why they are called ``Hamiltonian'' HDG methods. Because of this, when combined with symplectic time-marching methods, the resulting methods display an energy that does not drift in time. We provide a single analysis for several of these methods by exploiting the fact that they only differ by the choice of the approximation spaces and the stabilization functions. We also introduce a new way of discretizing the static Maxwell's equations in order to define the initial condition in a manner consistent with our technique of analysis. Finally, we present numerical tests to validate our theoretical orders of convergence and to explore the convergence properties of the method in situations not covered by our analysis.
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    Stormer-Numerov HDG Methods for Acoustic Waves
    (2018) Cockburn, Bernardo; Zhixing Fu; Hungria, Allan; Liangyue Ji; Sánchez Uribe, Manuel; Sayas, Francisco Javier
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    Supercloseness of Primal-Dual Galerkin Approximations for Second Order Elliptic Problems
    (2018) Cockburn, Bernardo; Sánchez Uribe, Manuel; Chunguang Xiong
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    Symplectic Hamiltonian finite element methods for electromagnetics
    (2022) Sanchez, Manuel A.; Du, Shukai; Cockburn, Bernardo; Nguyen, Ngoc-Cuong; Peraire, Jaime
    We present several high-order accurate finite element methods for the Maxwell's equations which provide time-invariant, non-drifting approximations to the total electric and magnetic charges, and to the total energy. We devise these methods by taking advantage of the Hamiltonian structures of the Maxwell's equations as follows. First, we introduce spatial discretizations of the Maxwell's equations using mixed finite element, discontinuous Galerkin, and hybridizable discontinuous Galerkin methods to obtain a semi-discrete system of equations which display discrete versions of the Hamiltonian structure of the Maxwell's equations. Then we discretize the resulting semi-discrete system in time by using a symplectic integrator. This ensures the conservation properties of the fully discrete system of equations. For the Symplectic Hamiltonian HDG method, we present numerical experiments which confirm its optimal orders of convergence for all variables and its conservation properties for the total linear and angular momenta, as well as the total energy. Finally, we discuss the extension of our results to other boundary conditions and to numerical schemes defined by different weak formulations.(c) 2022 Elsevier B.V. All rights reserved.
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    Symplectic Hamiltonian finite element methods for linear elastodynamics
    (2021) Sanchez, Manuel A.; Cockburn, Bernardo; Nguyen, Ngoc-Cuong; Peraire, Jaime
    We present a class of high-order finite element methods that can conserve the linear and angular momenta as well as the energy for the equations of linear elastodynamics. These methods are devised by exploiting and preserving the Hamiltonian structure of the equations of linear elastodynamics. We show that several mixed finite element, discontinuous Galerkin, and hybridizable discontinuous Galerkin (HDG) methods belong to this class. We discretize the semidiscrete Hamiltonian system in time by using a symplectic integrator in order to ensure the symplectic properties of the resulting methods, which are called symplectic Hamiltonian finite element methods. For a particular semidiscrete HDG method, we obtain optimal error estimates and present, for the symplectic Hamiltonian HDG method, numerical experiments that confirm its optimal orders of convergence for all variables as well as its conservation properties. (C) 2021 Elsevier B.V. All rights reserved.